We’ve introduced groups as an abstract structure that contains a set of elements together with a binary operation, whereas some specific group axioms have to be satisfied. On the contrary to a group GG, a ring RR contains two mathematical operations: an addition ++ and a multiplication \cdot. A ring is defined as an abstract structure that maintains an abelian group structure under the addition operation but not necessarily under multiplication.

Definition of a ring

A set of elements RR together with both operations

+ :R×RR,(a,b)a+b, and  :R×RR,(a,b)ab\begin{aligned} &+ \space : R \times R \rightarrow R,(a, b) \mapsto a+b, \quad \text { and } \\ &\cdot \space: R \times R \rightarrow R,(a, b) \mapsto a \cdot b \end{aligned}

is called a ring, if the following properties are fulfilled:

  • R1: RR is an abelian group under addition (+)(+).
  • R2: The multiplication \cdot is associative.
  • R3: The multiplication ()(\cdot) is distributive with respect to the addition (+)(+), i.e., for all a,b,cRa, b, c \in R :

    a(b+c)=ab+ac and (a+b)c=ac+bc.a \cdot(b+c)=a \cdot b+a \cdot c \text { and }(a+b) \cdot c=a \cdot c+b \cdot c.

The neutral element 0R0 \in R of the addition is called the zero elements of RR.

Note: We usually write (R,+,)(R,+, \cdot) to RR. The requirements for multiplication are weaker than the ones for addition because the multiplication operation doesn’t have to be necessarily commutative. There’s also no requirement for the existence of a multiplicative neutral element 11.

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